Learn More
and Applied Analysis 3 The following sharp lower power mean bounds for 1/3 G a, b 2/3 H a, b , 2/3 G a, b 1/3 H a, b , and P a, b can be found in 4, 6 : 1 3 G a, b 2 3 H a, b > M−2/3 a, b , 2 3 G a, b 1 3 H a, b > M−1/3 a, b , P a, b > Mlog 2/ logπ a, b 1.8 for all a, b > 0 with a/ b. The purpose of this paper is to answer the question: for α ∈ 0, 1 , what(More)
We find the greatest value α and the least value β such that the double inequality αT (a,b) + (1 − α)G(a,b) < A(a,b) < β T (a,b) + (1 − β)G(a,b) holds for all a,b > 0 with a = b. Here T (a,b) , G(a,b) , and A(a,b) denote the Seiffert, geometric, and arithmetic means of two positive numbers a and b , respectively. An optimal double inequality between(More)
For k ∈ 0, ∞ , the power-type Heron meanHk a, b and the Seiffert mean T a, b of two positive real numbers a and b are defined by Hk a, b a ab k/2 b /3 1/k , k / 0; Hk a, b √ ab, k 0 and T a, b a − b /2 arctan a − b / a b , a/ b; T a, b a, a b, respectively. In this paper, we find the greatest value p and the least value q such that the double inequalityHp(More)
In this paper, we find the largest value α and least value β such that the double inequality L α (a,b) < M(a,b) < L β (a,b) holds for all a,b > 0 with a = b. Here, M(a,b) and L p (a,b) are the Neuman-Sándor and p-th generalized logarithmic means of a and b , respectively. Optimal combinations bounds of root-square and arithmetic means for Toader mean, An(More)